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@ARTICLE{Ehrhardt:611791,
      author       = {Ehrhardt, Matthias J. and Kuger, Lorenz and Schönlieb,
                      Carola-Bibiane},
      title        = {{P}roximal {L}angevin {S}ampling with {I}nexact {P}roximal
                      {M}apping},
      journal      = {SIAM journal on imaging sciences},
      volume       = {17},
      number       = {3},
      issn         = {1936-4954},
      address      = {Philadelphia, Pa.},
      publisher    = {SIAM},
      reportid     = {PUBDB-2024-05071},
      pages        = {1729-1760},
      year         = {2024},
      abstract     = {In order to solve tasks like uncertainty quantification or
                      hypothesis tests in Bayesian imaging inverse problems, we
                      often have to draw samples from the arising posterior
                      distribution.For the usually log-concave but
                      high-dimensional posteriors, Markov chain Monte Carlo
                      methods based on time discretizations of Langevin diffusion
                      are a popular tool. If the potential defining the
                      distribution is non-smooth, these discretizations are
                      usually of an implicit form leading to Langevin sampling
                      algorithms that require the evaluation of proximal
                      operators. For some of the potentials relevant in imaging
                      problems this is only possible approximately using an
                      iterative scheme. We investigate the behaviour of a proximal
                      Langevin algorithm under the presence of errors in the
                      evaluation of proximal mappings. We generalize existing
                      non-asymptotic and asymptotic convergence results of the
                      exact algorithm to our inexact setting and quantify the bias
                      between the target and the algorithm's stationary
                      distribution due to the errors. We show that the additional
                      bias stays bounded for bounded errors and converges to zero
                      for decaying errors in a strongly convex setting. We apply
                      the inexact algorithm to sample numerically from the
                      posterior of typical imaging inverse problems in which we
                      can only approximate the proximal operator by an iterative
                      scheme and validate our theoretical convergence results.},
      cin          = {FS-CI},
      ddc          = {510},
      cid          = {I:(DE-H253)FS-CI-20230420},
      pnm          = {623 - Data Management and Analysis (POF4-623)},
      pid          = {G:(DE-HGF)POF4-623},
      experiment   = {EXP:(DE-MLZ)NOSPEC-20140101},
      typ          = {PUB:(DE-HGF)16},
      eprint       = {2306.17737},
      howpublished = {arXiv:2306.17737},
      archivePrefix = {arXiv},
      SLACcitation = {$\%\%CITATION$ = $arXiv:2306.17737;\%\%$},
      UT           = {WOS:001343420300001},
      doi          = {10.1137/23M1593565},
      url          = {https://bib-pubdb1.desy.de/record/611791},
}